TITLE: What is the category of covariant and contravariant functors?
QUESTION [11 upvotes]: Let $\bf Cat'$ be the category that has as objects small categories $A, B...$, and as arrows functors $F:A\to B$ that are either covariant or contravariant. The identity on $A\in\bf Cat'$ is the usual identity functor; the composition of a covariant and a contravariant functor is contravariant and the composition of two contravariant functors is covariant. (Here, by a contravariant functor $F:A\to B$, I mean a mapping $F: obj A \to obj B$ and for any arrow $f:x\to y$ an arrow $Ff:Fy\to Fx$ with the usual axioms on identities and composition.)
I wonder which are the abstract properties of $\bf Cat'$.
Note for instance that in $\bf Cat'$ any category is isomorphic to its dual; so I suppose that it is not possible to recover the covariant arrows in an abstract way. Maybe the obvious functor $\bf Cat \to \bf Cat'$ has some universal property?

REPLY [5 votes]: You may also be interested in my paper Contravariance through enrichment, which shows that $\bf Cat'$ is enriched over a certain non-symmetric monoidal structure on $\bf Cat\times Cat$.  This allows distinguishing the covariant and contravariant functors while still keeping them both present in the structure.  Thus for instance we can distinguish the "contravariant isomorphism" $A\cong A^{\rm op}$ from a normal covariant isomorphism, and indeed use the former to characterize $A^{\rm op}$ as "the unique (up to covariant isomorphism) object contravariantly isomorphic to $A$", which is also a copower of $A$ in the sense of enriched category theory.
This structure is actually closely related to the construction in Simon's answer; the monoidal structure on $\bf Cat\times Cat$ is constructed using the $\mathbb{Z}/2\mathbb{Z}$ action on $\bf Cat$, and can be generalized to other group actions.
Also I should note that when I say $\bf Cat'$ "is" enriched over $\bf Cat\times Cat$, I don't mean in the precise sense that $\bf Cat'$ is the "underlying ordinary category" of an enriched category in the standard sense of enriched category theory; the underlying ordinary category of the enriched form of $\bf Cat'$ is actually just $\bf Cat$.  This is similar to the situation of group actions, where a category enriched over $G$-sets is a category with $G$-actions on its hom-sets, but the "underlying ordinary category" takes the fixed-point sets of those hom-sets (rather than, for instance, forgetting the $G$-actions).